[Inner Product Spaces]
Definition: Let (V,F) be a vector space. An inner product is a map of the form ⟨⋅,⋅⟩:V×V→F such that for all x,y,z∈V and c∈F the following axioms hold:
(P1) ⟨x,y⟩=⟨y,x⟩​ (conjugate symmetry)
(P2) a. ⟨x+y,z⟩=⟨x,z⟩+⟨y,z⟩ (linearity in the first argument)
b. ⟨cx,y⟩=c⟨x,y⟩ (homogeneity in the first argument)
(P3) ⟨x,x⟩≥0 and ⟨x,x⟩=0 iff x=0v​ (positive definiteness)
fill missing parts
Theorem: "Cauchy-Schwarz Inequality" Let (V,F) be a vector space with an inner product ⟨⋅,⋅⟩. Then for all x,y∈V the following inequality holds:
∣⟨x,y⟩∣2≤⟨x,x⟩⟨y,y⟩
Proof: Let x,y∈V and c∈F.
-
0​≤⟨x−cy,x−cy⟩=⟨x,x⟩−⟨x,cy⟩−⟨cy,x⟩+⟨cy,cy⟩=⟨x,x⟩−c⟨x,y⟩−c⟨y,x⟩+∣c∣2⟨y,y⟩=⟨x,x⟩−c⟨x,y⟩−c⟨x,y⟩​+∣c∣2⟨y,y⟩=⟨x,x⟩−2Re(c⟨x,y⟩)+∣c∣2⟨y,y⟩​
#EE501 - Linear Systems Theory at METU